Download EX£RCIS£S - Model High School

Give the angle measure represented by each rotation.
10. 4.5 rotations counterclockwise
9. 2 rotations clockwise
Identify all angles that are coterminal with each angle. Then find one positive
angle and one negative angle that are coterminal with each angle.
11.22°
12. -170°
If each angle is in standard position, determine a coterminal angle that is
between D· and 360°. State the quadrant in which the terminal side lies.
13.453
14. -798°
0
Find the measure of the reference angle for each angle.
15.227°
16.-210°
17. Geography Earth rotates once on its axis approximately
every 24 hours. About how many degrees does a point on
the equator travel through in one hour? in one minute? in
one second?
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Change each measure to degrees, minutes, and seconds.
18. -16.75°
19. 168.35°
'20. -183.47°
21.286.88°
22. 27.465°
23. 246.876°
Write each measure as a decimal to the nearest thousandth.
24.23° 14' 30"
25. -14° 5' 20"
26. 233° 25' 15"
27. 173 24' 35"
28. -405° 16' 18"
29. 1002°30' 30"
0
Give the angle measure represented by each rotation.
30. 3 rotations clockwise
31. 2 rotations counterclockwise
32. 1.5 rotations counterclockwise
33. 7.5 rotations clockwise
34. 2.25rotations counterclockwise
35. 5.75 rotations clockwise
36. How many degrees are represented by 4 counterclockwise revolutions?
Identify all angles that are coterminal with each angle. Then find one positive
angle and one negative angle that are coterminal with each angle.
37.30°
38. -45°
39. 113°
40.211'
41. -199°
42. -305°
43. Determine the angle between 0° and 360°that is coterminal with all angles
represented by 3Ut + 360ko, where k is any integer.
44. Find the angle that is two counterclockwise rotations from 60°. Then find the
angle that is three clockwise rotations from 60·.
If each angle is in standard position, determine a coterminal angle that is
between D· and 360°. State the quadrant in which the "terminal side lies.
45.400°
46. -280°
47.940·
48.1059°
Lesson 5·1
49. -624°
50. -989°
Angles and Degree Meosure
281
51. In what quadrant is the terminal side of a 1275 angle located?
0
Find the measure of the reference angle for each angle.
52.327°
53.148°
55. -420
0
56. -197°
0
54.563
57. 1045°
58. Name four angles between 0° and 360° with a reference angle of 20°.
Applications
and Problem
Solving
59. Technology A computer's hard disk is spinning at 12.5 revolutions per second.
Through how many degrees does it travel in a second? in a minute?
60. Critical Thinking
Write an expression that represents
all quadrantal angles.
61. Biking During the winter, a competitive bike rider trains on a stationary bike.
Her trainer wants her to warm up for 5 to 10 minutes by pedaling slowly. Then
she Is to increase the pace to 95 revolutions per minute for 30 seconds. Through
how many degrees will a point on the outside of the tire travel during the 30
seconds of the faster pace?
62. Flywheels A high-performance composite flywheel rotor can spin anywhere
between 30,000 and 100,000 revolutions per minute. What is the range of
degrees through which the composite flywheel can travel in a minute? Write
your answer in scientific notation.
.
63. Astronomy On January 28, 1998, an x-ray satellite spotted a neutron star that
spins at a rate of 62 times per second. Through how many degrees does this
neutron star rotate in a second~ in a minute? in an hour? in a day?
64. Critical Thinking Write an expression that represents any angle that is
coterminal with a 25° angle, a 145 angle, and a 265 angle.
0
0
65. Aviation The locations of two airports
are indicated on the map.
a. Write the latitude and longitude of
the Hancock County-Bar Harbor
airport in Bar Harbor, Maine, as
degrees, minutes, and seconds.
b. Write the latitude and longitude of
the Key West International Airport in
Key West, Florida, as a decimal to
the nearest thousandth.
Hancock CountyBar Harbor Airport
north latitude 44.4499°
west longitude 68.2616°
Key West International Airport
north latitude 24 °33'32"
west longitude 81 °45'34.4"
66. Entertainment
A tower restaurant in Sydney, Australia, is 300 meters above
sea level and provides a 360 panoramic view of the city as it rotates every 70
minutes. A tower restaurant in San Antonio, Texas, is 750 feet tall. It revolves at
a rate of one revolution per hour.
0
a. In a day, how many more revolutions does the restaurant
make than the one in Sydney?
in San Antonio
b. In a week, how many more degrees does a speck of dirt on the window of the
restaurant in San Antonio revolve than a speck of dirt on the window of the
restaurant in Sydney?
282
Chapter 5 The Trigonometric Functions
...
Guided Pradice
5. Find the values of the sine, cosine, and tangent
for LT.
T
6. If sin (}=
7. It cot
(J
LJ~m
V
17'In.
i, find csc e.
= 1.5, find tan e.
Q
8. Find the values of the six trigonometric
ratios for LP.
.
6ft~
S
P
9. Physics You may have polarized sunglasses that eliminate glare by polarizing
the light. When light is polarized, all of the waves are traveling in parallel planes.
Suppose vertically polarized light with intensity 10 strikes a polarized filter with
its axis at an angle of e with the vertical. The intensity of the transmitted light
e are
It and
of~.
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if.
e=
related by the equation cos
If
(J
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XERCISES
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Find the values of the sine, cosine, and tangent for each LA.
10.
8:~
11.
C
~m
A
C
B
12.
B
12 in.
~
C
A
B
5ft
y
13. The slope of a line is the ratio of the change of
y to the change of x. Name the trigonometric
ratio of (}that equals the slope of line m.
_________
JIchange
f+-change in x-+j
In
y
x
e = 3'1 find cot
14. If tan
16. If sec (} =
18. If cot
(J
15. If sin
8.
t. find cos 8.
= 0.75. find tan e.
19. If cos
(J
= 0.125, find sec
e.
ratios for each LR.
21.
R
t. find esc 8.
17. If esc (}= 2.5, find sin 9.
Find the values of the six trigonometric
20.
e=
R
14cm~
22.
v7i~9in.
T
S
~
S
38 mm
T
s~
R
23. If tan
.(
(J
= 1.3. what is the value of cot (90
Chapter 5 The Trigonometric Functions
0
-
O)?
24. Use a calculator to determine the value of each trigonometric
a. sin 52° 47'
b. cos 79° IS'
d. cot 36° (Hint: Tangent and cotangent
Graphing
Cakulator
•
ratio.
c. tan 88° 22' 45"
have a reciprocal relationship.)
25. Use the table function on a graphing calculator to complete
the table. Round
values to three decimal places .
8
72°
74·
sin
cos
0.951
0.961
76·
78°
82·
80°
84·
86°
88·
0.309
e approaches
90°?
b. What value does cos 0 approach as 0 approaches
90°?
a. What value does sin () approach as
26. Use the table function on a graphing calculator to complete the table. Round
values to three decimal places.
8
sin
cos
tan
18°
0.309
l6·
14°
10°
8°
4·
6°
2·
0.276
0.951
e approach
cos e approach
a. What value does sin
b. What value does
c. What value does tan
Applications
and Problem
Solving
12°
e approach
e approaches
as e approaches
0°?
as ()approaches
0°?
as
0°?
27. Physics Suppose a ray of light passes from air to Lucite. The measure of the
angle of incidence is 45°, and the measure of an angle of refraction is 27" 55'.
Use Snell's Law, which is stated in the application at the beginning of the lesson,
to find the index of refraction for Lucite.
28. Critical Thinking
The sine of an acute LR of a right triangle is
values of the other trigonometric ratios for this angle.
t. Find the
When rounding a curve, the acute angle e that a
runner's body makes with the vertical is called the angle of
29. Track
2
incline. It is described by the equation tan
e = E.....,
where
gr
v is
the velocity of the runner, g is the acceleration due to gravity,
and r is the radius of the track. The acceleration due to gravity
is a constant 9.8 meters per second squared. Suppose the
radius of the track is 15.5 meters.
a. What is the runner's velocity if the angle of incline is 11°?
b. Find the runner's velocity if the angle of incline is 13°.
c.
What is the runner's velocity if the angle of incline is 15°?
d. Should a runner increase or decrease her velocity to
increase his or her angle of incline?
Uhf
. e =
se t e act th at sin
.
.
SIde adJacent.
.
cos e = hypo t enuse to wnte an expression
sin e and cos O.
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30 . Crltlca
Th·In kiIng
side
h opposite an d
ypotenuse
f
() .
Lesson 5-2
or tan
In
terms
f
0
Trigonometric Rotios in Right Triangles
289
CHECK
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Read and study the lesson to answer each question.
1. Explain why esc 180° is undefined.
2. Show that the value of sin () increases as
decreases as 8 goes from 90° to 180°.
.
3. Confirm that cot
=
£J
cos
-'-D'
SIl1
e goes
from 0° to 90 and then
0
e
!7
4. Maih 1fJWU«11 Draw a unit circle. Use the drawing to complete the chart below
that indicates the sign of the trigonometric functions in each quadrant,
tan a or cot a
Guided Practice
Use the unit circle to find each value,
5. tan 180°
6. sec (-90°)
Use the unit circle to find the values of the six trigonometric
each angle.
7,30°
functions for
8,225°
Find the values of the six trigonometric functions for angle 6 in standard position
if a point with the given coordinates lies on its terminal side.
10. (-6,6)
9. (3, 4)
Suppose 6 is an angle in standard position whose terminal side lies in the given
quadrant. For each function, find the values of the remaining five trigonometric
functions for 6.
1
11. tan () = -1; Quadrant IV
12. cos () = -2; Quadrant II
The distance around Earth along a given latitude can be found
using the formula C = 21Tr cos L, where r is the radius of Earth and L is the
latitude. The radius of Earth is approximately 3960 miles. Describe the distances
along the latitudes as you go from 0° at the equator to 90° at the poles.
1'3, Map Skills
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Use the unit circle to find each value.
14. sin 90°
15. tan 360
16. cot (-180°)
17. esc 270°
18, cos (-270°)
19. sec 180°
0
20. Find two values of
21. If cos
29",
.~
e = 0, what
Chapter 5 The Trigonometric Functions
8
for which sin () = 0,
is sec 8?
__ _
~_
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Use the unit circle to find the values of the six trigonometric
each angle.
22.45°
23.150
24.315
0
26.330
0
28. Find cot (-45°).
0
27.420
0
25.210
0
functions for
.
29. Find esc 390
0
•
Find the values of the six trigonometric functions for angle 8 in standard position
if a point with the given coordinates lies on its terminal side.
30. (-4, -3)
31. (-6,6)
32. (2,0)
33. (1, -8)
34. (5, -3)
35. (-8, IS)
36. The terminal side of one angle in standard position contains the point with
coordinates (S, -6). The terminal side of another angle in standard position
contains the point with coordinates (-5,6). Compare the sines of these angles.
37. If sin () < 0, where would the terminal side of the angle be located?
Suppose (} is an angle in standard position whose terminal side lies in the given
quadrant. For each function, find the values of the remaining five trigonometric
functions for 8.
12
38. cos () = -}3; Quadrant III
40. sin
(J =
42. sec
e=
-i;
Quadrant IV
v'i Quadrant
44.1f esc () = -2 and
Applications
and Problem
Solving
(j
IV
= 2; Quadrant II
39. esc
(j
41. tan
e = 2; Quadrant
43. cot
e=
lies in Quadrant Ill,find tan
I
1; Quadrant III
e.
If you ignore friction, the
amount of time required for a box
to slide down an inclined plane is
45. Physics
J .!a
gsm
cos
9' where
a is the horizontal
distance defined by the inclined plane,
g is the acceleration due to gravity, and
is the angle of the inclined plane. For
what values of () is the expression
undefined?
e
46. Critical Thinking
For each statement,
describe k.
a. tan (k . 90°) = 0
b. sec (k . 90°) is undefined.
47. Physics
.
For polarized light, cos
e
Ii:
=
.
'1/ -;- ,where e is.the angle of the axis of the
o
polarized filter with the vertical.v, is the intensity of the transmitted light, and 10
is the intensity of the vertically-polarized light striking the filter. Under what
conditions would I, = Io?
48. Critical Thinking The terminal side of an angle e in standard position coincides
with the line y = -3x and lies in Quadrant II. Find the six trigonometric
functions of 8.
Lesson 5·3
Trigonometric Functions on the Unit Circle
297
3. Name the angle of elevation and the angle
of depression in the figure at the right.
Compare the measures of these angles.
Explain.
~_ -
4. Describe a way to use trigonometry to
determine the height of the building where
you live.
0
Solve each problem. Round to the nearest tenth.
Guided Practice
13 and A = 76°, find a.
B.lf B
=
=
7. If B
=
16° 45' and c
5. If b
26° and b
=
A
;~B
18, find c.
= 13, find a.
a
8. Geometry Each base angle of an isosceles triangle measures 55° 30'. Each of
the congruent sides is 10 centimeters long.
a. Find the altitude of the triangle.
b. What is the length of the base?
c. Find the area of the triangle.
The Ponce de Leon lighthouse in St. Augustine, Florida, is the second
tallest brick tower in the United States. It was built in 1887 and rises 175 feet
above sea level. How far from the shore is a motorboat if the angle of depression
from the top of the lighthouse is 13° IS'?
S. Boating
EXERCISES
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Solve each problem. Round to the nearest tenth.
A
10. ItA = 3r and b = 6, find a.
11. If c
.
= 16 and B = 670, find a.
._c
b
12. If B = 62° and c = 24, find b.
13. If A
=
29° and a
14.Ifa
=
17.3andB
15. If b = 33.2 and B
= 4.6, find c.
=
=
7r,findc.
B
61°, find a.
=
= 16° 55' and c =
1B.lf B = 49° 13' and b
10, find a.
17. If A
13.7, find a.
20. Geometry. The apothem of a regular pentagon
is 10.8 centimeters.
a. Find the radius of the circumscribed circle.
b. What is the length of a side of the pentagon?
c. Find the perimeter of the pentagon.
C
..g
Exercises 10-18
p~12
18. If a = 22.3 and B = 47° 18', find c.
19. Find h, n, m, and p. Round to the nearest tenth.
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~
j.-m
n--+-j
Exercise 19
21. Geometry Eacli base angle of an isosceles triangle measures 42° 30'. The base
is 14.6 meters long.
a. Find the length of a leg of the triangle.
b. Find the altitude of the triangle.
c. What is the area of the triangle?
302
Chapter 5 The Trigonometric Functions
)
22. Geometry A regular hexagon is inscribed in a circle with diameter
6.4 centimeters.
a. What is the apothem of the hexagon?
b. Find the length of a side of the hexagon.
c. Find the perimeter of the hexagon.
d. The area of a regular polygon equals one half times the perimeter
of the
polygon times the apothem. Find the area of the polygon.
Applications
and Problem
Solving
23. Engineering The escalator at St. Petersburg Metro in Russia has a vertical rise
of 195.8 feet. If the angle of elevation of the escalator is 10° 21' 36", find the
length of the escalator.
24. Critical Thinking Write a formula for the volume of
the regular pyramid at the right in terms of a and s the
length of each. side of the base.
a
s
25. Fire Fighting The longest truck-mounted ladder used by the Dallas Fire
Department is 108 feet long and consists of four hydraulic sections. Gerald
Travis, aerial expert for the department, indicates that the optimum operating
angle of this ladder is 60°. The fire fighters find they need to reach the roof of an
84-foot burning butlding. Assume the ladder is mounted 8 feet above the ground.
a. Draw a labeled diagram of the situation.
b. How far from the building should the base of the ladder be placed to achieve
the optimum operating angle?
c. How far should the ladder be extended to reach the roof?
26. Aviation When a 757 passenger jet begins its
descent to the Ronald Reagan International
Airport in Washington, D.C., it is 3900 feet from
the ground. Its angle of descent is 6°.
Distance
Traveled
~
3900ft~~rt
Grourld Distance
a. What is the plane's ground distance to
the airport?
b. How far must the plane fly to reach the runway?
The Cape Hatteras lighthouse on the North Carolina coast was
built in 1870 and rises 208 feet above sea level. From the top of the lighthouse,
the lighthouse keeper observes a yacht and a barge along the same line of sight.
The angle of depression for the yacht is 20 and the angle of depression for the
barge is 12° 30'. For safety purposes, the keeper thinks that the two sea vessels
should be at least 300 feet apart. If they are less than 300 feet, she plans to
sound the horn. How far apart are these vessels? Does the keeper have to sound
the horn?
27. Boat Safety
0
,
G
28. Critical Thinking Derive two formulas for the
length of the altitude a of the triangle shown at
the right, given that b, S, and 8 are known.
Justify each of the steps you take in your
reasoning.
Lesson 54
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Applying
Trigonometric
Functions
303
Solve each equation if 0° esx
Guided Practice
5. cosx
:S
360°.
1
="2
-\13
6.tanx=
-3-
Evaluate each expression. Assume that all angles are in Quadrant I.
7. sin (Sin-1
8. tan' (COS-1
~)
%)
Solve each problem. Round to the nearest tenth.
9. If r
10. If
.~~
= 7 and s = 10, find R.
T
r = 12 and t = 20, find S.
R
s
Solve each triangle described, given the triangle at the
right. Round to the nearest tenth if necessary.
11.8 = 78°, a = 41
12. a
= 11,
13.A
=
b
= 21
32<, c
=
13
14. National Monuments In 1906,
Teddy Roosevelt designated Devils
Tower National Monument in
northeast Wyoming as the first
national monument in the United
States. The tower rises 1280 feet
above the valley of the Bell Fourche
River.
a. If the shadow of the tower is
2100 feet long at a certain time,
find the angle of elevation of
the sun.
Devils Tower National Monument
b. How long is the shadow when the angle of elevation of the sun is 38°?
c. If a person at the top of Devils Tower sees a hiker at an angle of depression of
65°, how far is the hiker from the base of Devils Tower?
,_ _ .r
Practice
EXERCISES
_,....".,_,.
_
r
_...
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=-..,
-
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Solve each equation if 0°
15. sinx
J
=
_-
.~
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16. tan x
1
19'.
18. cos x = 0
->
_-v
-
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""'''''''
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_-
360°.
=
_:v'3
17.cos x
v'2
smx = --2-
21. Name four' angles whose sine equals
\13
= -2-
20. tan x =-1
t.
Evaluate each expression. Assume that all angles are in Quadrant I.
22. cos (arccos
t)
25. esc (arcsin 1)
23. tan (tan-1
~)
26. tan (cos-1
1
24. sec ( cos-1
53)
27. cos (sin-l
Lesson 5-5
- ~~
--------
----
f)
f)
Solving Right Triangles
309
Solve each problem. Round to the nearest tenth.
28. If n
= 15 and m = 9, find N.
M
29. If m
=
= 14, find M
p = 30, find M.
n
30. If
8 and p
n = 22 and
31. If m
= 14.3 and n = 18.8, find N.
32. If p = 17.1 and m
33. If m
=
P~--~N
.m
7.2, find N.
= 32.5 and p = 54.7, findM.
If the legs of a right triangle are 24 centimeters and 18 centimeters
long, find the measures of the acute angles.
34. Geometry
The base of an isosceles triangle is 14 inches long. Its height is
8 inches. Find the measure of each angle of the triangle.
35. Geometry
Solve each triangle described, given the triangle at the right. Round to the
nearest tenth, if necessary.
36. a
=
21, C
=
30
38. B = 47°, b = 12.5
40.
C
42. B
Applications
d Problem
o;olving .
37. A
39.
= 35°, b = 8
a = 3.8, b = 4.2
= 9.5,b = 3.7
41. a =13.3,A
= 33°, C = 15.2
43. c
= SIS
K
a
b
A~B
12
= 9.8,A = 14°
44. Railways The steepest railway in the world is the Katoomba Scenic
Railway In Australia. The passenger car is pulled up the mountain by twin
steel cables. It travels along the track 1020 feet to obtain a change in altitude
of 647 feet.
a. Find the angle of elevation of the railway.
b. How far does the car travel in a horizontal direction?
45. Critical Thinking
a. sln "! 2.4567
Explain why each expression Is Impossible.
b. sec "! 0.5239
c. cos " (-3.4728)
46. Basketball The rim of a basketball
hoop is 10 feet above the ground.
The free-throw line is 15 feet from the
basket rim. If the eyes of a basketball
player are 6 feet above the ground,
what is the angle of elevation of
the player's line of sight when
shooting a free throw to the rim of
the basket?
Several years ago, a
section on 1-75near Cincinnati, Ohio,
had rise of 8 meters per 100 meters of horizontal distance. However, there
were numerous accidents involving large trucks on this section of highway.
Civil engineers decided to reconstruct the highway so that there Is only a rise
of 5 meters per 100 meters of horizontal distance.
a. Find the original angle of elevation.
b. Find the new angle of elevation.
47. Road Safety
a
31 0
Chapter 5 The Trigonometric Functions
CHECK
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Read and study the lesson to answer each question.
1. Show that the Law of Sines is true for a 30"-60° right triangle.
2. Draw and label a triangle that has a unique solution and can be solved using
the Law of Sines.
3. Write a formula for the area of
parallelogram HlXYZ in terms of a, b, and X.
WaX
Z
,// /?
a
4. You Declde Roderick says that triangle MNP
has a unique solution if M, N, and m are known.
Jane disagrees. She says that a triangle has a
unique solution if M, N, and p are known. Who
is correct? Explain.
Y
M
N ~
P
m
Guided Practice . Solve each triangle. Round to the nearest tenth.
= 40°, B = 59°, c = 14
7. If B = 17° 55', C = 98° 15', and a
S.A
6. a
= 17, find
=
=
8.6, A
27.3°, B
=:
55.9°
=
14
c.
Find the area of each triangle. Round to the nearest tenth.
S.A
=
78°, b
=
=
14, c
9. A
12
=
B
0
22
,
=
105°, b
10. Baseball Refer to the application at the beginning of the lesson. How far is the
baseball fan from the pitcher's mound?
.
;+
._"
•
Practice
2_::'~
E
~
X£RCISES
,,-_ ......L.~
... _~
_,;
-~..I:f,.-c_----
~.. '--....
~
-,-_.,.,.'--_......,...-~--
••-It.:::::
.. ~
••
-
~""';
'--~"""''''_7E
-",._
-
.... ~_.~-+
Solve each triangle. Round to the nearest tenth.
11. A
= 40°, C
13. b = 12,A
15. a
=
8.2, B
= 70°, a = 20
= 25°, B
= 35°
=
1S. Whatis a if b
11, B
=
51° 30', and
=
=:
100
14. A
=
65°, B
0
,
C= 50°,
= 50°,
16. C = 19.3,A
= 24.8°, C = 61.3°
17. ItA = 37° 20', B
12. B
C
=
=
C
C
= 30
=
12
39° 15', C= 64° 45'
125, find b.
29° 34', and C
=
23° 48'?
Find the area of each triangle. Round to the nearest tenth.
19.A = 28°, b
= 14, C = 9
21.A
=
15°, B = 113°, b = 7
23. B
=
42.8°, a
= 12.7, C = 5.8
=
22. b =
24. a =
20. a
5, B
= 3T,
146.2, C
19.2, A
C
=
84°
= 209.3, A = 62.20
=
53.8°, C
=
65.4°
25. Geometry The adjacent sides of a parallelogram measure 14 centimeters and
20 centimeters, and one angle measures 57°. Find the area of the parallelogram.
26. Geometry A regular pentagon is inscribed in a circle whose radius measures
9 inches. Find the area of the pentagon.
27. Geometry A regular octagon is inscribed in a Circle with radius of 5 feet. Find
the area of the octagon.
'3)._ Chapter
5 The Trigonometric Functions
...
Applications
and Problem
Solving
28. Landscaping A landscaper wants to plant begonias .along the edges of a
triangular plot of land in Winton Woods Park. Two of the 'angles of the triangle
measure 95 and 40°. The side between these two angles is 80 feet long.
0
a. Find the measure of the third angle.
b. Find the length of the other two sides of the triangle.
c. What is the perimeter of this triangular plot of land?
For 6MNP and MST, LM
the Law of Sines to show b.MNP - MST.
29. Critical Thinking
== LR, LN == LS. and LP == L T. Use
30. Architecture
The center of the
Pentagon in Arlington, Virginia,
is a courtyard in the shape of a
regular pentagon. The pentagon
could be inscribed in a circle
with radius of 300 feet. Find the
area of.the courtyard.
31. Ballooning A hot air balloon is flying above Groveburg. To the left side of the
balloon, the balloonist measures the angle of depression to the Groveburg
soccer fields to be 20 IS'. To the right side of the balloon, the balloonist
measures the angle of depression to the high school football field to be 62 30'.
The distance between the two athletic complexes is 4 miles.
0
0
a. Find the distance from the balloon to the soccer fields.
b. What Is the distance from the balloon to the football field?
32. Cable Cars· The Duquesne Incline is a
cable car in Pittsburgh, Pennsylvania, which
transports passengers up and down a
mountain. The track used by the cable car
has an angle of elevation of 30°. The angle of
elevation to the top of the track from a point
that is horizontally 100 feet from the base of
the track is about 26.8°. Find the length of the track.
100 tt
33. Air Travel In order to avoid a storm, a pilot starts the flight 13°off course.
After flying 80 miles in this direction, the pilot turns the plane to head toward
the destination. The angle formed by the course of the plane during the first
part of the flight and the course during the second part of the flight is 160°.
a. What is the distance of the flight?
h. Find the distance of a direct flight to the destination.
34. Architecture
An architect is designing an overhang
above a sliding glass door. During the heat of the. '..
summer, the architect wants the overhang to prevent the
rays of the sun from striking the glass at noon. The
overhang has an angle of depression of 55" and starts
13 feet above the ground. If the angle of elevation of the
sun during this time is 63°, how long should the architect
make the overhang?
Lesson 5-6
The Law of Sines
317
. (\
"
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Communicating
Mathematics
CHECK
,
.
FOR
._ ...
._
.,
UNDERSTANDING
-.
• <_. ".
, '''''
"'r
I
c""
t ~ -'
'"
...~~
•
,,'
.".
....~,
Read and study the lesson to answer each question.
1. Describe the conditions where the Law of Sines indicates that a triangle cannot
exist.
2. Draw two triangles where A = 30 a = 6, and b == lO. Calculate and label the
degree measure of each angle rounded to the nearest tenth.
0
,
3. Write the steps needed to solve a triangle if A
Guided Practice
= 120
0
,
a = 28, and b
= 17.
Determine the number of possible solutions for each triangle.
4.A = 113°, a
=
IS, b
5. B = 44°, a
8
=
23, b
=
=
none. Round to the
Find all solutions for each triangle. If no solutions exist, write
nearest tenth.
6. C
= 17°, a = 10, c = 11
8.A
= 38°, b =
10, a
= 140°, b = 10, a = 3
9. C = 130°, C = 17, b = 5
7. A
8
=
12
10. Communications
Avertical radio tower is located on the top of a hill that has
an angle of elevation of 10°. A 70-foot guy wire is attached to the tower 45 feet
above the hill.
a. Make a drawing to illustrate the situation.
b. What angle does the guy wire make with the side of the hill?
. c. How far from the base of the tower is the guy wire anchored to the hill?
. '---..,<r~.".
E
,r
....
Practice
~
"
X£RCISES
,-,'
... -.=
_-
;.r,:'----
1.
... rIl--
~
_'.-
-
•
-_ ~
'~"".="
....
_..... """' ...
"'...._
13. B
=
=
=
= 19
61°, a = 12, b = 8
57°, a
11, b
15. C = 100°, a = 18,
17. If A
=
65°, a
=
C
= 15
55. and b
12. A
= 30°, a = 13, C = 26
14. A
=
=
16. B
=
=
a=
58°, C
94°, b
37°.
32. b
c_ '"
18. a = 6, b :::;;
8, A
= 4, b = 8
B = 60°. C =
22. A
40°,
24. B
=
36°, b = 19.
26. A
=
76°,
28. B
= 40°, b = 42,
20
= 30
a = 5, b = 20
C
= 60
23. a = 14. b = 12, B
25. A
=
27. C
= 47°, a =
29. b
=
107.2°, a
40. a
measurements of the triangle.
ts.s cm
~
Functions
none. Round to the
=
10.
= 90°
17.2. c
C
=
12.2
= 16
= 32, A = 125.3°
30. Copy the triangle at the right and label all
The Trigonometric
27
= 26, b = 29, A = 58°
21. C = 70°, C = 24, a = 25
30·, a
C
=
19. a
= 150°
=
=
20. A
= 17
57. how many possible values are there for B?
Find all solutions for each triangle. If no solutions exist, write
nearest tenth.
Chapter 5
;(-
Determine the number of possible solutions for each triangle.
11. A
32..
~-;=-
.~.
A
21.7 om
31. Find.the perimeter of each
b = 20 and A = 29°.
of the two noncongruent
triangles where
.
a = 15,
32. There are two non congruent triangles where B = 55°, a = 15, and b = 13.
Find the measures of the angles of the triangle with the greater perimeter.
Applications
and Problem
Solving
33. Gears An engineer designed three gears as shown
at the right. What is the measure of 8?
r= 15 em
~.~
. .
~bIICO~'O"
34. Critical Thinking If b = 14 and A = 30°,
determine the possible values of a for each
situation.
a. The triangle has no solutions.
b. The triangle has one solution.
c. The triangle has two solutions.
35. Architecture
The original height of the
Leaning Tower of Pisa was 184t feet. At a
distance of 140 feet from the base of the tower,
the angle of elevation from the ground to the
top of the tower is 59°. How far is the tower
leaning from the original vertical position?
36. Navigation
The captain of the Coast Guard
Cutter Pendant plans to sail to a port that is
450 miles away and 12° east of north. The
captain first sails the ship due north to check a
buoy. He then turns the ship and sails 316 miles to the port.
a. In what direction should the captain turn the ship to arrive at
the port?
Port
b. How many hours will it take to arrive at the turning point if
the captain chooses a speed of 23 miles per hour?
c. Instead of the plan above, the captain decides to sail
200 miles north, turn through an angle of 20° east of north,
and then sail along a straight course. Will the ship reach the
port by following this plan?
37. Communicatio.ns
A satellite is orbiting Earth
every 2 hours. The satellite is directly over a
tracking station which has Its antenna aimed 45°
above the horizon. The satellite is orbiting
1240 miles above Earth, and the radius of Earth
is about 3960 miles. How long ago did the
satellite pass through the beam of the antenna?
(Hint: First calculate e.)
Lesson 5-7
Beam
from --7-=----ry~::......._
. Antenna
I
j
I
I
\
\
'\
" ' .... _---.,..
The Ambiguous Case for the Law of Sines
325
~.
10. Softball In slow-pitch softbaJl, the diamond is a square
that is 65 feet on each side. The distance between the
pitcher's mound and home plate is 50 feet. How far does
the pitcher have to throw the softball from the pitcher's
mound to third base to stop a player who is trying to
steal third base?
-----t(
150 f
65 It
:
I
o
EXERCISES
,-,l-
._"
Practice
_
••
~"",,'
~ • .r• .!_
'\.
'"
I
.',~
-...__
.. ..". ••
,
•
~<
.'
.....
_
•
"....
Solve each triangle. Round to the nearest
11. b
=
7, c
=
10, A
=
13. a
=
4, b
=
5, C
15. a
= 11.4, b = 13.7, c = 12.2
=
51 °
7
H.;;_..
~O:'"io2!LC
....,..".'.
..
•
.....
--...-.
tenth.
12. a
=
5, b
=
14. a
=
16,
16. C
= 79.3°, a = 21.5, b = 13
C
6, c
=
= 12,
7
B = 63°
17. The sides of a triangle measure 14.9 centimeters, 23.8 centimeters, and
36.9 centimeters. Find the measure of the angle with the least measure.
18. Geometry Two sides of a parallelogram measure 60 centimeters and
40 centimeters. If one angle of the parallelogram measures 132°, find the length
of each diagonal.
Graphing
Calculator
Programs
Find the area of each triangle.
For a graphing
19. a
4, b
21.
20, b
calculator
program that
determines
the area of a
triangle, given
the lengths
of
all sides of the
triangle,
visit
www.amc.
glencoe.com
=
a =
23. a
= 174,
Round to the nearest
20. a = 17, b = 13, C = 19
= 6, C = 8
= 30, C =
b = 138, C
tenth.
40
=
188
=
22. a
=
24. a
= 11.5, b =
33, b
51,
25. Geometry The lengths of two sides of a parallelogram
30 inches. One angle measures 120°.
C
= 42
13.7,
C
=
12.2
are 48 inches and
a. Find the length of the longer diagonaJ.
48 in.
b. Find the area of the parallelogram.
26. Geometry The side of a rhombus is 15 centimeters
longer diagonal is 24.6 centimeters.
long, and the length of its
a. Find the area of the rhombus.
b. Find the measures of the angles of the rhombus.
ApplIcations
and Problem
Solving
27. Baseball
In baseball. dead
center field is the farthest point
in the outfield on the straight line
through home plate and second base.
The distance between consecutive
bases is 90 feet. In Wrigley Field in
Chicago, dead center field is 400 feet
from home plate. How far is dead center
field from first base?
28. Critical Thinking
The lengths of the
sides of a triangle are 74 feet, 38 feet, and 88 feet. What is the length of the
altitude drawn to the longest side?
•
Lesson 5-8
The Low of Cosines
33 1
,
__
.
29. Air Travel The distance between Miami and Orlando is about 220 miles. A pilot
flying from Miami to Orlando starts the flight 10° off course to avoid a storm.
I
a. After flying in this direction for 100 miles, how far is the plane from Orlando?
b. If the pilot adjusts his course after 100 miles, how much farther is the flight
than a direct route?
30. Critical Thinking
Find the area of the
pentagon at the right.
201.5 It
A soccer player is standing 35 feet from one post of the goal and 40 feet
from the other post. Another soccer player is standing 30 feet from one post of
the same goal and 20 feet from the other post. If the goal is 24 feet wide, which
player has a greater angle to make a shot on goal?
31. Soccer
32. Air TraHic Control A
757 passenger jet and
a 737 passenger jet
are on their final
approaches to San
Diego International
Airport.
15,000 ft
.-
..
"~
........;
....
.:~
-"
."_.
a. The 757 is 20,000 feet from the ground, and the angle of
depression to the tower is 6°. Find the distance between
the 757 and the tower.
b. The 737 is 15,000 feet from the ground. and the angle of
depression to the tower is 3°. What is the distance
between the 737 and the tower?
c. How far apart are the jets?
Mixed Review
33. Determine the number of possible solutions for MBCifA
a ::= 17. (Lesson 5-7)
= 63.2°, b = 18 and
The San Jacinto Column in Texas is 570 feet tall and, at a particular
time, casts a shadow 700 feet long. Find the angle of elevation to the sun at that
time. (Lesson 5-5)
34. Landmarks
,
r
35. Find the reference angle for -775°. (Lesson 5-1)
II
36. Find the value of k so that the remainder of (.x3 - 7x2
(x - 3) is O. (Lesson 4-3)
-
kx
+ 6) divided by
e:t
37. Find the slope of the line through polnts at (2t, t) and (5t, 5t). (Lesson 1-3)
38. SAT/ACT Practice
6
A 8x
r
332
Find an expression equivalent to
6
B 64x
Chapter 5 Ih« Trigonometric Funclions
y3
C 6_x5
r
0
s.xs
r
2xS
Ey4
I ixtra
i
j
.
Pradice
See p. A35.
.
.
,
-;t'
J
I
':, ~
'~\
CHAPTER
",L'·,.~
OBJECTIVES
REVIEW EXERCISES
Find the values of the six trigonometric
functions for angle (J in standard position if
a point with coordinates (3, 4) lies on its
terminal side.
=
v'x2
sin fJ
tan
+ y2
= )!_ =
r
=
vI32+42
sIn
•
5
(J
= - =y
4
r
5
II
= -yX
=4
11
Find the values of the six trigonometric
functions for each angle 8 in standard position
if a point with the given coordinates lies on its
terminal side.
27. (3,3)
28. (-5,12)
29. (8, -2)
30. (-2,0)
32. (-5, -9)
31.(4,5)
33. (-4,4)
34. (5,0)
3
=-
r
5
3
r
V2s or 5
cos () = csc
x
=
x
i5
e =)!_x = i3
sec()=-=-
STUDY GUIDE AND ASSESSMENT
AND EXAMPLES
Lesson 5-3 Find the values of the six
trigonometric functions of an angle In standard
posttton given a point on its terminal side.
r
5 •
3
Suppose 8 is an angle in standard position
whose terminal side lies in the given quadrant.
For each function. find the values of the
remaining five trigonometric functions for 8.
35. cos () =
3
-"8;
Quadrant
II
36. tan (J = 3; Quadrant ill
..........................................................................................................................................................................................................................................
Lesson 5-4 Use trigonometry to find the
measures of the sides of right triangles.
r
Solve each problem. Round to the nearest
tenth.
A
Refer to MRe at the right. If A = 25° and
b = 12, find c.
cos A
cos 25
0
L}
=!!...c
12
=-
8
c
a
c
12
c=-=-cos 25°
c = 13.2
37. If B = 42° and c = 15, find b.
38. If A = 38° and Q = 24, find c.
39. If B = 67° and b = 24, find Q.
,
:
Lesson 5-5
Solve each equation if 0° s x s 360°.
Solve right triangles.
~i::
~o;nd
roo.
Q
10
A =
LJ
= 9, find A.
sin A = ~
A
b
C
sm " _2_
A =64.2°
10
B
.
,
a
C
40. tan
e=
~
41. cos
e=
-1
Refer to i":!.ABC at the left. Solve each triangle
described. Round to the nearest tenth if
necessary.
42. B = 49°, a = 16
43. b = 15, c = 20
44.A=64 ,c=28
0
.
:
Chapter 5 Study Guide and Assessment
337
!
.
'j~,1
" .
.'
5•
CHAPTER
STUDY GUIDE AND aSSESSMENT
_,_:::-~..
...._
~
~
REVIEW EXERCISES
OBJECTIVES AND EXAMPLES
Change each measure to degrees, minutes,
and seconds.
11.57.15°
12. -17.125°
Lesson 5-1 Identify angles that are coterminal
with a given angle.
-::::::1:.'
_:.
I
• ...____
II a 585° angle is in standard position,
determine a coterminal angle that is
between 0° and 360°. State the quadrant in
which the terminal side lies.
If each angle is in standard position, determine
a coterminal angle that is between 0° and
360°. State the quadrant in which the terminal
side lies.
13.860
14. 1146°
15. -156°
16. 998
17. -300°
18. 1072°
19. 654
20. -8320
First, determine the number of complete
rotations (k) by dividing 585 by 360.
0
0
585
360 = 1.625
Use a + 360ko to find the value of a.
a + 360(1)" = 585
a = 225°
The coterminal angle (a) is 225°. Its
terminal side lies in the third quadrant.
0
0
..................................................................................................................
Find the measure of the reference angle for
each angle.
21. -284°
22. 592
0
u
H
Lesson 5-2 Find the values of trigonometric
ratios for acute angles of right triangles.
Find the values of the six trigonometric
ratios for LM
23. Find the values
of the sine, cosine,
and tangent for LA.
.
A
~9<m
B
15cm
C
Find the values of the six trigonometric
functions for each LM.
24.
. M =5"4
sm
cosM = 1.
5
tanM
4
="3
. M
Sill
side opposite
hypotenuse
=
cos M =.
side adjacent
hypotenuse
N
si 'de Q dirj(1cent
25.
cscM=-
5
4
5
secM= -3
3
cotM= -4
esc
hypotenuse
M =,
,
side opposite
sec
M
=
cot M -
~ypotenuse
side adjacent
side adjacent
'd
'
SI e opposite
~
336
M
12 m
- side opposite
tan M -
8m
P
M
P
~
Win.
26. If sec 8 =
•
Chapter 5 The Trigonometric Functions
~>
.111
SKILLS AND CONC£PTS
+
" ~
N
t, find cos 8.
h
_
.
OBJECTIVES
REVIEW
AND EXAMPLES
Find the area of a triangle if the
measures of two sides and the included angle or
the measures of two angles and a side are given.
Lesson 5-6
Find the area of MBC if a
C = 54°.
= 6, b = 4, and
Draw a diagram.
K= tab sin C
= -}(6)(4) sin 54°
K
K = 9.708203932
B
6
C
EXERCISES
Solve each triangle. Round to the nearest
tenth.
45. B = 70°, C = 58°, a = 84
46. c = 8, C = 49°, B = 57°
Find the area of each triangle. Round to the
nearest tenth.
47. A = 20°, a = 19, C = 64°
4B. b = 24,A = 56°,B = 78°
49. b = 65.5, c = 89.4, A = 58.2°
50. B = 22.6°, a = 18.4, c = 6.7
f
I
The area of MBC
is about 9.7 square units .
..........................................................................................................................................................................................................................................
Solve triangles by using the Law of
Lesson 5-7
Sines.
In MBC, if A
find a.
= 51°, C = 32°, and c =
Draw a diagram.
a
sinA
=
=
38.7°, a
=
=
203
~
Abe
(sin 51°)18
sin 32·
a
26.4
:
.
,
Lesson 5-8
Solve triangles by using the Law of
Cosines.
•
172, c
52. a = 12, b = 19, A = 57"
53. A = 29°, a = 12, c = 15
54. A = 45', a = 83, b = 79
a
18
a=
r=
51. A
~B
c
sin C
a
18
--~=-sin 51'
sin 32°
18,
Find all solutions for each triangle. If no
solutions exist, write none. Round to the
nearest tenth.
In 6ABC, if A
= 63°,
b
=
20, and c
=
14,
Solve each triangle. Round to the nearest
tenth.
55. A = 51°, b = 40, c = 45
56. B
finda.
~B
Draw a diagram.
14
= 19°, a = 51, c = 61
57. a = 11, b
a
58. B
= 13,
C
= ?O
= 24', a = 42, c = 6.5
63·
A
20
C
+ c2 ~ 2bc cos A
a2 = 202 + 142 - 2(20)(14) cos 63°
a2
=
b2
a2 = 341.77
a = 18.5
..........................................................................................................................................................................................................................................
338
Chapter 5
The Trigonometric Functions